3/3 Classical transversality methods in SFT
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3/3 Classical transversality methods in SFT

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36

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

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English

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Abstract 
There are easy examples showing that classical transversality methods cannot always succeed for multiply covered holomorphic curves, but the situation is not hopeless. In this talk I will describe two approaches that sometimes lead to interesting results: (1) analytic perturbation theory, and (2) splitting the normal CauchyRiemann operator of a curve along irreducible representations of its automorphism group. Both were pioneered by Taubes in his work on the Gromov invariant and SeibergWitten theory in the 1990's, and I will illustrate them by sketching two proofs that the multiply covered holomorphic tori counted by the Gromov invariant are regular for generic J. If time permits, I will discuss some ideas as to how both methods can be applied more generally.

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04:26
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21:24
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29:53
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47:13
Logical constant
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54:31
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INTEGRAL
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57:00
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Infinity
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Fundamental theorem of algebra
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and so a top and we and you have to do and thus find many costs I think it's fair to say all the results of talking about are pretty well considered standard and approves of described are also using wireless standard methods the title of the many costs has lowered classical classical concerts yourselves what I mean by that is that any result that you can expressed in the form for generic domain independent almost complex structures days we have Kansas salary for some particular constitutional curves so that's allowed to include some methods which cannot be called standard and that's what we to talk about today In particular because everything I talked about so far with 1 minor exception when I talked about automatic has a salary and everything else makes the restriction that we're only talking about somewhere injective curves which of course is a big problem if you want to define the gene variants such as government era and To a large extent you cannot really hope for classical it has a solid results to be true in the generality that you would need to define those theories about deuce already illustrated this briefly in all reiterate that at that moment but there are situations when those results can hold when you don't need to go to much more general framework such as the Polly fold theory In order to define everything if you do get a chance to study for a honest along with the Kurds it can make your life easier because the cochairman equation is much easier to handle and carries some sort of natural geometric information that you don't necessarily have In whatever perturbed equation you're going to solve the more general framework so so sometimes it just requires much more originality in the way of thinking To get waiting at the meeting the give an illustration so for today and I'm going to have to depart a little bit from the title of the courses and not really talk about their safety so I'm not going to talk about punctured holes the cursor for disclosed all of Kurds say segments in prime time will be closed surfaces almost all of what I'm going to say can very likely be generalized to punch a hole Martha curves but that's workinprogress so I'm not really going to touch upon it and besides if you general ideas I'll just tell you about the things that I so a multiple cover looks like this I have to take a 2 1 home I the branch cover from 1 Ramon service to another suitcase greater than 1 Sigma Prime is the domain of Sunday whole curve into an almost complex manifold W. J. capital J and then the composition of the 2 homers maps gives me the multiple colors assumes the is somewhere injected new is now a key full coverage Hoffa occurs and it'll be helpful to note that there is a relationship between the on the characteristics of these 2 domains as they both figuratively index formula this is the 1 Hermits formula that says minus by the characteristic of Sigma plus of the degree of the coveted times on the characteristics of the prime equals this quantity that I like to call the increased 5 because that's literally algebraic count of the zeros of the fight on other words as the number of branch points countered with with other waters of branching a number of
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critical points the fight and inspires said all those killed positively services integer size greater than so we have this as a constraint relating the drawings if you've never seen the formula before you can just think of it this way deified is a section of a certain complex line bundle which you can easily right out and you can compute C 1 of that line bundle the answer is the lefthand side of this form some of the key to the news you can also draw some pictures you can you can triangulate things and dropped serve refreshments looks maybe you prefer but it's very quick thinking of it as the 1 this is the favorite the formula that I remember so that's right down the index farmers that we get from Ramon rock so the index of the sub W assume to M dimensional Real to ndimensional the index of the by retaining the virtual dimensions of the modular space on parameterize Hoffa curves that the lives so that's not literally the fragile minutes of linear especially monopoly but it is bye basically what I explained before it is that plus the dimension of the relevant tasteless the that's the actual dimension of marvelous space if stresses holds and the formula please and minus 3 times on the characteristics of the domain plus 2 1st has evaluated the movement across the how does that relate to the index of you the same formula with different domain now remind her Hurwitz tells me I can rewrite the characteristic of cinema as k times electrical circuits in the prime minister's count of branch points as per se 1 of users Kate see 1 of these sites to sites that and now I see sitting inside his firm Cape times the index of the because it's an minus 3 times already stretches Surkov prime was 1 of the over here I just have an extra term so again OK times the index of the minus and minus 3 times discounted branch so in particular to be a little bit more commentary on what can go wrong with how the study for multiple covers suppose that the underlying some injective Curtis index 0 so that's a generically a rigid isolated object in his marvelous space and for generic today it's also going to be stable and a small perturbations J which means there's no way you can get rid of the multiple covered curve since there will always be a perturbed multiple cover of perturb injective curve but the index of youth Is there going to be and like likeminded Simitis 3 times this nonnegative counter branch points which can easily be negative at least before and dimension 8 or upward the covers on branched that's going to be a negative number right dimension upwards that's precisely where we stopped being able to assume their syntactic manifold is something positive so there we ran into trouble this is likely to be a negative number in general but we see that we cannot perturbed that curve away the situation is actually even worse if you think about that what kind of curves we already know must exist in a neighborhood of you is not just you itself there's the other nearby breast covers of the Cindy and those ,comma nontrivial modular space in general so the actual dimension of the moduli spaces all articles near you and when I say actual dimensions I'm not making the assumption that system manifold for anything it might not be you might not even be some form of old but it does contain a smooth order fold which I can identify very clearly namely I the space of branched covers system overseen a primer on the same degree so the Soviet the dimension of the space of an hour case 1 branch covers Margalo we privatization and that's very well understood what I say is as just another modular space of Hallmark the curves living in dimension to if you like and you can compute the dimension from the remark formula as usual the answer is 2 times the number of GM plants Anderson a geometric interpretation the newly and this is my next sentence just interrupted and this is a geometric interpretation of this because they're given a branch cover you can find other branch covers nearby by moving around the positions of the branch points in the and those are not equivalent up to reprivatization you different vent covers and this is precisely the number of parameters you see but doing that so this this is basically classical so we see that number is never going to match the actual index of you on last possibly the covers on branch for what were for any dimension to that someone we don't really care about mentioned too and so so here it's fair to say chances Saudi's generally not plausible for the multiple cover I do want to make another observation about this so again the antics of the so generically by resigning after possibly perturbing J I'm allowed to assume that some were injected into servers are immersed in case there's why is that so I've discussed this with a couple of people since my last the talk a discussion session whatever it was that there is an exercise posed about using automatic 100 solidly to prove that in a fourdimensional some practical board is generically an index 1 hole or sixcylinder will always be regular and part of what you have to do to prove that it is observed that whatever kind of hold cylinder that 1 covers of the multiple cover it is also going to be a sufficiently low index so that you can assume it's arrest and then you can apply so there's general factor which I don't have time to talk about it in earnest 1 can show that if you take your usual modular space Adam Oct . 2 it's so that increases the dimension space by 2 but now constraint that mark .period by asking for the derivatives the 1st derivatives Iraq to vanish but that mark .period so In general that decreases the dimension largest expect to end so the
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upshot of that is for generic the space of the whole Norfolk somewhere injective presented not immersed may be considered to be of CO dimensions 2 minutes to end compared to the large amount of space so that means if Lanexa 0 I can assume that that's the synonymous Kansas empty therefore this one's immersed some not wasting more about that was just accepted for announces it J homomorphic fact analogous to the standard differential topological facts that you can perturb smooth maps to be given the right special conditions so no we have this no that means let's look at the normal bundle I've got a new theory of why this was a descendant of 1 yet because you have to work out the details they go ahead and try and let me know how to comply so the normal bundle is not the generalize but the usual definition of normal but I think that's what it see what is that generalize normal bundle of you it's very easy to describe to you is not embarrassed because there can be branch points of the critical point of view but of course and generalized moral bundle you just think it will be the pullback at this moment when the so that's to the times that number over there see 1 of the bank's characteristics in prime time member I talked last time about restricting the linear Koshy remind operated to the generalized normal bundle that also gives you could be 1 type operator on a bundle of bring 1 lower it's fragile indexes given directly by the remote rock formula let's see what it is to do your end is the restriction to the normal bundle and rock tells me well the rank of this bundles and minus 1 as a complex fumbled sighs eerie characteristic of the domain plus 2 won the generalized normal bundled so everyone who it's tells me what that 1st term ends we have this extra turn accounting commander of critical points and then to see 1 and you have just written appear it's there we go there I'm missing something at saying accusing her of the year and the will of course be events you and that's why that's the case but I think I think this is a clear society were roundabout way than I planned presently but that OK times In minus 1 of the characteristics minus another to or the characteristics um plus twice see 1 of the Ben Linus and I just want to find out about this is fine because this number in brackets 0 0 because they assumed that these indexes are 0 as the index formula just left with reminds last month at Zeev defect so no 1 knows about this is unlike via the index view that I wrote down just looking at the normal operator there some predictable pattern this number is always mom positive which means it's conceivable that this normal operator might actually always be injected and that's something geometrically meaningful if it's true so I'm going to actually status as a conjecture about a year ago at this time I was calling it a theorem but then there was discovered in that proved so this is a conjecture that says so Virginia Jr all multiple colors you somewhere injectors index 0 curves the has normal country operator injected which has a nice consequence if it's true what actually implies is all of the other Kurds in the neighborhood of you are precisely the ones you already know about the just the other multiple covers the so I think this this result would give you a precise description of the moduli space occurs near you as having exactly this dimension of the space of bread that would be the intention right so the the intention of a result like this would be to prove something like the Gulf lob a conviction I felt that the reason why this kind of thing is supposed to be interesting is that it means in certain settings if you want to compute Gromov Whitman variants you really only have to understand the somewhere injective curves and the rest of it of course the multiple covers are not regular in the usual sense so you have to do some kind of perturbation if you want to actually count them but there's a standard way of doing this with in homogeneous federations because you an equation and you can predict the count the you'll get because you see the entire modular space in terms of the space branch covers it has obstruction bundling compute the underclass class of obstruction on leveled against the this has my qualities and beloved the answer to your question is yes and the main thing I want to say about it right now is that since it is generally negative the operator can be injected and what I really want to explain is how you interpret the fact that operators injectors it's or if the kernel these operators trivial think about it like this the
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following is a scenario you don't want you don't want to have a sequence of curves there are have different images from you converging to you OK my claim is that all the sequences of curves that can converge to you are of the form the compose with some other branch cup so they all have the same now if you have a sequence of cars with different images converging toward you you can do this trick where you you look at that there's Kurds living in the normal bundle of you and we scale that normal bundle so that as the curves approach you re scale so that you see them not actually approaching but staying in some bounded subset of the normal bundle apply Groleau compactness to that that sequence is going to converge to some generally mobile home off a curve which will have some component that you can interpret it as something in either in the kernel of this operator or in the kernel of of some a related operator corresponding to a abreast cover of lesser degree so if you know that these kernels are all trivial that precludes the scenario and so it was really that the spaces branched covers space and use the rest comes V is an open subset Of the whole much this is all speculation about and the argument that I just described can be made 4 years ago but of course the conjecture only conjecture and we don't know if this is true so I do want to talk about the special case of it we do know yes you would think that what is going on strike I'd have to think about that a little bit I certainly could come up with examples given enough time where I am particularly pleased there's hardly any conditions here the main condition just the simple courtesy next year so in situations where that's not true right now the possibility of a sequence of a simple cars converging to something mostly covered is always something you have to worry about you usually have to make some effort to avoid it and I'm gonna talk about that a little bit but I don't think it's an example of this phenomenon and it's much nicer than that so His actual which is 1 of the cases of this conjecture and this is in a joint paper that I wrote with Chris Gallagher last year I'm from going to make a statement a little bit unnecessarily more complicated than they need just to illustrate how different it is from the results we talked about so far so let's say fix and open subset you in there closed manifold em "quotation mark selected manifold dimension to an and thanks also became almost complex structure J. fixed then I will say there exists a call major but subsets J. living as you'd expect in the space of all JayZ or only detained and matched 6 outside of this subset such that for all Jason this class Paul covers What's new of somewhere injected index 0 curves called the contained in the perturbation domain I regularly so remember talk about unrest covers then the disaster scenario I described with this index relations doesn't happen massacres were initially because you implies index you for the next Eagles 0 behind use also 0 so in that case regularity in the usual sense is plausible numerically I think generic data actually happens at least if we're looking at curves contained entirely inside this perturbation only that's 1 major difference With the films I explained earlier I'm not saying we have regularly for all the curves that intersect or have haven't .period mapping into the perturbation domain but they have to be contained in it entirely so we'll see why that seems to be necessary I don't I don't know if that condition can be dropped as an adult know dropped the although you can insulating you to be the whole but if you want to just do perturbations in some subset and then you have to restrict use of that the use of the money will use closed I you and is closed therefore you have contact with this I mean I I could also allow him to be not compact and then I wouldn't have to be part of the U has compact cost that's a good question so I'm not going to explain the in quite this level of generality but I'm going to explain a case of which is somewhat older than our resolve but somehow very badly known a badly understood so in the case and it was too so it's just to mention for 10 where most domains are Taurus and the underlying symbol :colon was actually invented so this is what do so alluded to was question moments ago this is done by Childs in 1996 paper in the Journal of the the truth there is kind of hidden that's when I when I went back to last nite figure out where it was it took me while and it's very sketchy but somehow it's the the ideas in it extremely potent so the cost this means that we're getting regularly for multiple covered almost tore the covering inventory tells this because his definition of the role of invariant actually counted those things and I did it without doing abstract 4 divisions that did it for generic so In the situation on the Taurus there some convenient
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things the fact that the has index 0 means that it's normal bundle has to be trivial so I can write normal bundling the couldn't identified with trivial bundle T complex line bundle similarly and a coterie of an operator fire right in those terms just looks like the usual operated plus 7 0 Florida so here by bar operator I mean literally got partial bite at specified times partial lately acting on public valued functions an aide is safe see out from the chorus to the space of real linear an amorphous and the sea that's what is 0 return looks like and then the pulled back normal operators literally the pullback of that so the more if for you is going to be the standard D plus this is your furniture a compose with the cover fire so just a few general points before I really get into the argument it suffices to show the following suffices to show that for all day the 1st of the team class and thinking about indeed the jail was the embedding of the Taurus into this perturbation domain I knew it was the compose with some cover fire fighters necessarily unrest since it's a tourist covering Taurus then I want to show that I can J 2 J. prime such that the is still the prime home effect but the normal operated for you becomes an isomorphism they defined with respect to the return of the complex structure the price b claims its definition is a simple you just you take ordinary coach Ramon operator which is defined in the pullback tangent bundle you restricted to sections of the normal bundle and now you get some section of a larger about movies don't want but also has a projection 2 a corresponding a normal part that is that it takes you to some sections of the bundle homebody segment too you start W T whatever it's called there's no projection and that is what he sees a homebody circuit to normal so you just composer culture operators that North to the fastest time about music middle of the there is another nicely to think about this so there's an alternative way of describing a neighborhood of in its modelers face that I didn't talk about if the crime is the worst specifically so so if the car is a modest 1 way of describing all the other curves nearby it's actually write this little bits of music .period all the other Kurds nearby indeed soon to be on the floor of X U H 4 7 8 section of the moment now In fact that's going to hit a unique parameterization of every nearby curve in the modular space but you don't get to choose what complex structure you have in the domain at all so you have to do that is not actually look for a nearby maps of this form that solved the coach Ramon equation with respect to some specific Jaber is rather nearest also still look for nearby maps like this who's tangent spaces a adjacent invariant so that automatically you can pull that back to some complex structure the diminutive get to prescribe so that's another way of seeing all the whole occurs in the vicinity of the swimmers and well you can write down some nonlinear operators does that free irritation is essentially this normal ?question mark that also tells you why it happens that the Phantom index of normal operators the same as the dimension of the moduli space in numerous case this is the time to do would write perturbations in the tent direction Mark meaningful for studying this modular space because this giving you we dramatizations of the sinker OK Son of everyone had a chance in intend to think about why what I just said here true suffices to show given occurrence and the multiple cover you can charge a 2 1 that makes that specific multiple cover regular in the sense of the normal operating licenses so this is an exercise using the trick that I explained last time thank you you can exhaust the space you're interested in with the accountable Union of Compaq subsets in this case finite subsets so as long as you're able to to achieve Francis salary for each subsets then you can find some set of Jesus accountable intersection opened and sets that does everything so that's all and say this is a version of the Telstra lady from there to the result we really want the other thing is going to say is we can reduce this problem to something that's that's really only involving linear coterie among operators on an abstract vector bundle because I can say no all 0 4 returns a prime scissors auditors just some scenes in the function valued in the space of building maps and see we can find the J prime such that the equals jails in the tangent directions on the current the thirtysomething years is when I was calling me generalized tantamount before literally that just means it the image of the differential the and we can also Seiji chronicles J outside some neighborhood of the image of the but the normal operator for expressed In trivialization expressed with respect to but returned almost complex structure is was a prime so I'm only saying here give me any a perturbed cushion on operator you want In the space of all reeling at 41 operators I can find a perturbed jaded realizes that for and this is not terribly deep tree in the embedded case when these invented this is fairly easy to prove it's just a matter of choosing the the normal 1st derivative of your 4 terms J in the
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right way to produce resident 1 can also do it in numerous states and that's a bit more painful and I'm not going to talk about that OK so let's also take this is given and just look at a problem involving coaches on operators of line bundles In the middle of the nite but it would all the image a DVD another the tangent spaces to the curve on there the other questions so his acclaimed and I'm even going to label this 1 improbable claim because when I 1st saw this fear Childs's 96 people I had no idea why I should believe this is true and I'm still not sure I can explain to you why you should leave this is true but I can prove it let's suppose Dean of the firm Dubai plus 8 is a cozy line operator on the trivial bundle of energy to trivia line bundle and the is a bundle map on that bundle which I'm going to assume is a complex substantially bundle isomorphism so the 2 key properties complex and healing and it's bundle so given that the bundle is trivial it's obvious that you can do this this is more or less equivalent to saying these he acts on a vector data certified by some complex life function data the Times complex conjugate associated with this complex valued function data is assumed to be nowhere 0 it's maddening to see stock 2 that's the assumption I want to just mention quickly if I were not working on the Taurus but with an eye on branched cover With more general domains segments of a kind the assumption that the index of the simple Kyrgyz 0 allows me again to do this then I would be able to find a complex and bundle isomorphism between the relevant bundles even another nontrivial that actually equivalent to the fact that the index of the simple curve 0 so that's that something so that needed to check now the statement will be that I can define perturbed operator the talent scout plus the real parameter times this extra money will not be the traders 0 the attack that His inasmuch as Faulks how do numbers outside some discreet subsets so this is 1 way of perturbing and pursuing operator that might not be an isomorphism and making it into a nice amorphous In particular where you need to notice about this perturbation Is it doesn't care at all about symmetry required expert from B to be a complex and unionized workers and I don't require to be anything else is my bundle is a pullback of some bundled as defined by a simple curve but I have been looking actually at the normal operator animals covered curve I can do a perturbation like this a lot just by changing J. along with simple curved like pull that back to the multiple cover my perturbation of my operator is going to be invariant now under DEC transformations of the cover this claim does not care West symmetry messes up the usual argument on this but this is the reason why I need some rejected unusual arguments symmetry ruins this this sorts mail argument this is impervious to that so I need to convince system "quotation mark there is a compact perturbation the operator so have this doesn't change the index basically all of at home operators in In this talk have index 0 so the isomorphisms are they have both Colonel and yet and was a I'm not it would be nice if we have this that means I can find so if I have play given Jay and my given cover which is maybe not regular I can find a perturbed to J that perturbed In this way and therefore make sacrifices so 1st steps this 1 from the family of operators the tower is injected which of course since its index 0 implies it's nice once and for all Taos efficiently Lawrence so this step is interesting because I'm quite convinced that is argument couldn't have come from somebody who was mainly as inflected topologist it had to come from gage tourists In particularly if you're familiar with tells us work relating to Gromov invariants inside wooden invariants you'll notice a parallel here there's there's something that tells dozens of wooden 3 when you write down sided with equations with this perturbation term depending on real parameter and you can prove that for topological reasons if you make that parameter very large you don't have any solutions we do have solutions but there are converging in the sense of current something like that so this is going to be a much easier version of that it's an argument that I've never seen anywhere else in Timbuktu topology so let's say we think about but I don't I haven't really specified all what what spaces and working with researchers to make my life easier and easier work 2 silvers spaces and say that operators reformate want all too so I suppose I have some 1 0 elements Ada In H 1 and Torres and the idea is to operate on that with the tiles and look to North square the details of some of 2 terms which means I can expand this all too apparent and if the terms I have Decatur also squared plus tell squared times fell to arm of the perturbation the which I'm reading writing here has been at times it Obama solidified their Nevada a few months spread and that is across that I'm assuming
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that my inner product is realvalued because the the Linear operators is reeling enough complex in so I can use
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a commissioning in a product on the real complex line bundle but actually the real part so this is going to look like to tout the real part of the L to their product and they had a prime with Devina was right that out as the body plus data 3 so a few observations about this Of course this 1st nonnegative obviously these the 2nd term since I assumed that need the is a bundle isomorphism which means this map data is nowhere 0 I condemn this term from below by the Ulster normal data this is greater than could to some constant C 1 and C L 2 nor is it a spread and that's all I need to care about that too Over here I really have to terms in the crossterm look at the more harm was 1 1st the pairing of data in a bar with a header that also is clearly His absolute value is bounded above by another constant times people to nauseated square and I have to worry a little bit about there was the other part of the cross terms so I need to work a little bit estimate that properly so I much happier that term if it didn't involve a derivative of Ada you have all to pairing of something with something else that's a derivative what you do you can integrate by parts let's see the real power in a product made in a bar with device is literally the real power of the interval L the conjugate beta Baer has in Tennessee violator integrated ready to now user like role say grants interval of the Marvel think the lab. Times in minus the integral of people made about the kinds of data integrating Deveyra something of a close manifold give me 0 due to stocks or will probably be given reduced that to the fundamentals calculus that's 0 over here I can extend a little bit further and say minus real part and you got it bar plans theater and Spain a number still minors this is 1 of those arguments which you get 1 sign wrong you really dead Peter got decried media OK hopefully at signs right and I am missing an the idea that you that's important because I want this to be the same as that term but I had on the lefthand side so I can put this last year on the lefthand side so I have twice that equals this but so the thing and try to estimate actually equals minus onehalf real part the integral part of the club Peterborough Ada times Ada and I don't really have to care about the details of this anymore either I just want to say the absolute value of all this it is now less than or equal to some other constant times the ultimate on the square so I put that all together members the whole thing was that they were worried I am saying that if you take a complex valued function on the Taurus and integrate the bottom of whole Torres you always get 0 and then you can proved that using stocks there Q some of the most minute I mean it was a mail to product to start with but I read it as a product of complex numbers of I 2 conjugates of of the chancellor yeah so was a little bit this is this is an interval of a complex valued function that's expressed as a product of competent function time it equals that to this part right so I'm defining IOC prior to be the real part of this integral so the U.S. nuclear plants in the history of this this is emerging the usual 1 so he's a good questions in a kind of allude to the fact that 1 can do all this and much more general setting but it's because they would have had so this is not a uniquely lowdimensional phenomenon describing even going describing a lowdimensional proved you can do it in higher dimensions you can also do it more general domains just requires several extra steps that so some this detailed in L 2 months where it is now greater than or equal to some constant which I'm going to change the name of constancy 1 primetime TV squared minus another constancy to primetime style who tells squared tell time Els who normally data and that's the proof it is nontrivial then this cannot be 0 as long as has sufficiently large but the was the most I would like to call the 1st
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1 years because accounts squares 3 the remaining difficulty in this argument was that I had to get control over this term that has the derivative it and then relayed that to just the plain old tune on the back that integration parts for and that's also where I used hate you may not have noticed it so explicitly but that's where you assumption that by perturbation term is a complex and winner cultivation so specifically because it appears and there has been at times the conjugated leader if I didn't have it embarked on the upper left this argument would have worked as close as I can come to getting intuitive reason to believe that I'm sorry so In the context of the love the only thing we can do about it it whatever is convenient there is 1 do you believe that there is some constant accepted the ceremony will longer than before all right so I'm almost done but I have 1 more bit of magic to pull off to finish this and that's the analytic perturbation theory so I won't have too much time to How much time do actually have agreed and that's always answers in the management theory so what I'm about to say it can be done in the real analytic category but I don't want to because I'm a little bit allergic to the real analytic category so instead and work in the complex analytical category and complexity Sinai operator 1st my operators only real linear even those acting in a complex vector bundle so what I can do at the expense of having 2 complex structures in the picture instead of 1 he likened complexity fire the domain and the target of the operator and considered the canonical complex linear operator the extension of that to the complex's occasions so what's
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more precisely the Tel is a real linear operator found some space complex very H 1 functions to complex 2 functions let's take um actually to be more precise we can say this is a H 1 functions from S 1 2 2 and colleague are 2 instead of will help avoid some confusion coming up as aromatic complexity fired as I attitude of some of the best Canadians as 1 in my notes that's not of good so if we complexity 5 this becomes a model call detail see From Howard H 1 C which you can think of as well the tense a product of this silver space with which is equivalent to the space of 1 functions from tee to proceed isn't about that too corresponding space fell to see and its complex linked and what I can do now is allowed my parameter touted the complex instead of just now the map taking towel To the operator detail is a whole more perhaps From the sea To the space of Fred home index 0 operators complex linear from page 1 complex applied to Elkview complex flight which of course is an open subset of the space of bounded linear operators complex linear between those spaces so this is just a complex upstate I have a map from into that box space is easily seen to be differential With respect to the complex variable tout its effect is so that's a whole market wrapped into this Open subset of frontal index 0 operators so the last step is into the the following this as nonverbal operators so operators that not isn't sitting inside his face with friend home in it operators is what we call an analytic some variety the complex analytical some variety which means locally you can express it as the 0 sets of some hold Norfolk functions on that infinite dimensional so open subset an infinite dimensional complex fastpaced that function is valued in something fundamental Cianci effect so what we end up with is the set of all parameters tell conceived with the property that the complex fight operator details see is modernize what isn't 2 looks locally like 0 set up a whole wasn't function the CTC that's why that's so is discreet we already showed that it's not but it's not everything because for very large tells the operator is a metamorphosis so 1 November 2 0 set must be discreet adult and then of course you have to do you to think a little bit about relating the statement about the complexity occasion to the original real linear operator that's fine basically you can convince yourself that if power is real then the complex's site operators being an isomorphism implies that the real operators some of so I was going through this summer in fact I and say after that there's no time so if you want to know the proof of the Lemmer I tell you whenever the 5 which holds that no but not enough so let's have 1 you know the company uses the inside of an what happens yeah for other large complex parameters that are positively alliance the only thing you use of your name and in your house and this was when I haven't thought about their future questioned so you probably think the whole time and there is much more than the Europe it was the industry leader in Europe you on the company In the end of the year so OK my goal I had a specific normal cursory 1 operated for this mostly covered curve so this is what I think is abstract discussion at the end I'm calling just plain the and detail becomes just some perturbation of this monolog which corresponds to some specific choice of perturbation of your most complex structure in the neighborhood of the current so we do not reset the clock and the stock to president thank and